Mathematical model of an unbiased and random arrangement of elements in an inherently biased die

ABSTRACT

Systems and methods are disclosed herein to a die comprising a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, when a count (c i ) of each unique element in the unbiased set is equal to 1.

TECHNICAL FIELD

The present invention relates generally to gaming involving dice, and more specifically, to the arrangement of numerical elements on a biased die.

BACKGROUND

Dice are commonly used in amusement activities such as board games, role playing games, or gambling. A die typically is a shaped like cube, and each side of the die has a. different integer value printed in some way, such as using dots or a printed numeral. By rolling the die, each side has the same probability of facing up (⅙^(th) chance). If all of the sides have exactly the same probability of facing up, the die is called an unbiased die. A die can become biased by altering the weight distribution of a die or by labeling more than one side with the same numerical element. In either situation, one side of the die or one element labeled on the die may have a higher probability of being rolled, thus making the die biased.

While a die commonly has six sides, a die may have any number of sides. For example, role-playing games may use dice with more than six sides, such as a ten-sided die (decahedron), a twelve sided die (dodecahedron), and a twenty sided die (icosahedron). On any unbiased die with any number of faces (n=number of faces on a die), the faces may be labeled so that the numbers on opposite faces of the die add up to equal n+1. For example, on a six-sided die, 1 and 6, 2 and 5, and 3 and 4 may be placed on opposite sides of the die. The placement of the numbers in this manner assures that the numbers 1, 2, and 3 share a vertex.

A die can be modified so that certain numbers appear on more than one face of the die. For example, if a die has six sides, but only five elements, such that the number 5 appears on two sides of the die, then the die is biased toward the number 5 because the probability of rolling a 5 is ⅓, while the elements 1, 2, 3, and 4 each have a ⅙ probability of being rolled. No matter where the two 5 elements are placed, the die will be biased.

A biased die may be useful for certain games. For example, a sixteen-sided die (hexadecagon) could be used to play chess using luck rather than strategy. In this chess example, the die would have one side representing a king (K), one side representing a queen (Q), two sides representing bishops (B), two sides representing knights (Kn), two sides representing rooks (R), and 8 sides representing pawns (P). A serial order placement would be: K, B, Kn, R, P, B, Kn, R, P, P, P, P, P, P, P. While the die is in this chess example is inherently biased toward the pawn, the die is farther biased toward the pawn element in a serial order placement of the elements because the pawn elements are not evenly distributed about the faces of the hexadecagon die. In order to place the elements for this die in an even and unbiased distribution, elements of the same type (i.e., two pawn elements) are to be placed on opposite sides of the die. However, finding the exact opposite for all elements may be difficult as the number of faces on a die increases. Thus, a method of placing elements on an inherently biased die in an unbiased method is desired.

SUMMARY

The systems and methods described herein attempt to overcome the drawbacks discussed above by providing an inherently biased die with elements placed in an unbiased manner. A number of mathematical expressions that depend on the sum of the count of the total number of unique elements on the die provide guidance in placing the elements on the inherently biased die. The final placement of elements on the die provides an unbiased and even distribution of elements.

In one embodiment, a die comprises: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, when a count (c_(i)) of each unique element in the unbiased set is equal to 1.

In another embodiment, a die comprises: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing ((N/n)−c_(i)) elements from the biased set of elements after over (N−n)/n occurrences of an element from the unbiased set of elements, if a count of the elements in the unbiased set less the elements having an identical value as the elements in the biased set of elements is greater than a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set.

In another embodiment, a die comprises: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing ((N/n)−c_(i)) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements, if a count of the elements in the unbiased set less the elements having an identical value as the elements in the biased set of elements is less than or equal to a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set.

In another embodiment, a die comprises: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled. by placing ((N/n)−c_(i)) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements, when a count (c_(i)) of each unique element in the unbiased set is equal to n.

In another embodiment, a computer-implemented method of labeling elements on an inherently biased die having a plurality of faces (N), comprises: determining, by a computer, whether the count of each unique element (c_(i)) in an unbiased set of elements is greater than 1; applying, by a computer, a first placement means if the c_(i) of the unbiased set of elements is equal to one; determining, by a computer, whether the count of each unique element (c_(i)) in an unbiased set of elements is equal to a number of elements (n); applying, by a computer, a fourth placement means if the c_(i) of the unbiased set of elements is equal to n; determining, by a computer, whether a count of the elements in the unbiased set less the elements having an identical value as the elements in an biased set of elements (c1) is greater than a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set (c2); applying, by a computer, a second placement means if c1 is greater than c2; and applying, by a computer, a third placement means if c1 is less than or equal to c2.

Additional features and advantages of an embodiment will be set forth in the description which follows, and in part will be apparent from the description. The objectives and other advantages of the invention will be realized and attained by the structure particularly pointed out in the exemplary embodiments in the written description and claims hereof as well as the appended drawings.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings constitute a part of this specification and illustrate an embodiment of the invention and together with the specification, explain the invention.

FIG. 1 illustrates a twelve sided die with labeled sides according to an exemplary embodiment.

FIG. 2 illustrates an inherently biased twelve sided die with elements placed in a biased arrangement according to an exemplary embodiment.

FIG. 3 illustrates an inherently biased twelve sided die with elements placed in an unbiased arrangement according to an exemplary embodiment.

FIG. 4 illustrates faces of an inherently biased twenty sided die with elements placed in a biased arrangement according to an exemplary embodiment.

FIG. 5 illustrates faces of an inherently biased twenty sided die with elements placed in an unbiased arrangement according to an exemplary embodiment.

FIG. 6 illustrates the steps in the method of placing elements in an unbiased manner on an inherently biased die according to an exemplary embodiment.

FIGS. 7A-C illustrate examples where c_(i)=1 according to an exemplary embodiment.

FIGS. 8A-8C illustrate examples where c_(i)=2 according to an exemplary embodiment.

FIGS. 9A-C illustrate examples where c_(i)=3 according to an exemplary embodiment.

FIGS. 10A-C illustrate examples where c_(i)=4 according to an exemplary embodiment.

FIGS. 11A-C illustrate examples where c_(i)=5 according to an exemplary embodiment.

FIGS. 12A-C illustrate examples where c_(i)=6 according to an exemplary embodiment.

DETAILED DESCRIPTION

Reference will now be made in detail to the preferred embodiments, examples of which are illustrated in the accompanying drawings.

Placing elements on an inherently biased die in an unbiased manner may be accomplished by applying various techniques according to the exemplary embodiments described herein, such as the four exemplary placement techniques. The placement techniques applied to the placement of elements on a die use mathematical models implemented by a computer that depend on how may faces the die has, how many elements are to be included on the die, and how many elements are included in a set of unbiased elements and a set of biased elements. More specifically, the set of unbiased elements is the set of elements that all have the same probability of being rolled, The biased set of elements includes the elements that make the die biased. For example, if a six-sided die is biased and includes the set elements 1, 2, 3, 4, 5, and 5, the unbiased set of elements includes the elements 1, 2, 3, 4, and 5, whereas the biased set of elements includes the additional 5 that makes the die biased toward the number 5. These placement techniques will be illustrated below with the assistance of a number of examples and FIGS. 1-6.

The placement techniques included in the exemplary embodiments rely on a number of computer-calculated mathematical models using variables that include a number of sides on a die (N), a number of unique elements to be included on the sides of the die (n), and a count of each unique element (c_(i)). The number of unique elements included on the sides of the die (n) must be exactly divisible by the number of sides on the die (N)) and [N<=(2n+n)]. Further, the count of each unique element (c_(i)) is the number of times that each unique element appears on the die. Returning to the chess example above, the count (c_(i)) of the king and queen would be 1, the count (c_(i)) of the rooks, knights, and bishops (c_(i)) would be 2, and the count (c_(i)) of the pawns would be 8. The value of each integer element on the sides of the die may be represented by the variable i, where 1≦i≦n.

A first exemplary placement technique, which is carried out by a placement module executed by a processor and stored on a non-transitory computer readable medium, for unbiased placement of elements on an inherently biased die, is applied to the situation where c_(i)=1 for i=(1 to (n−1)) and c₁=(N−(n−1)) for i NOT=(1 to (n−1)). The expression i=(1 to (n−1)) mathematically represents the situation where the biased element is the highest numbered element. However, it should be noted that this is for illustration purposes only, and any element may be the biased element from 1 to n. To illustrate the first placement technique, a 12-sided die (N=12) having six unique elements (n=6) is taken as an example. The six unique elements are i=1, 2, 3, 4, 5, 6, and, c_(i)=1 for i=1, 2, 3, 4 5, and c_(i)=7 for i=6. In the first placement technique, an element from the unbiased set of elements is placed after every (N−n)/n occurrences of an element from the set of biased elements.

Applying the first placement technique to the example above, all of the elements on the die would be 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 6. The unbiased set would be 1, 2, 3, 4, 5, 6; and the biased set would be 6, 6, 6, 6, 6, 6. By having a computer apply the mathematical model used by the first placement technique, (N−n)/n=1, and thus, an element from the unbiased set is placed after every 1 occurrence of an element from the biased set. Thus, the placement of the elements on the die is in the order of 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6. Referring now to FIGS. 1-3, the placement of the elements on the die is illustrated. A computer may further decide how to place the elements once the proper mathematical models have been calculated and applied by the computer.

As shown in FIG. 1, the faces of the dodecahedron are numbered 1 through 12. FIG. 2 illustrates the serial placement of the elements on the dodecahedron. As shown in FIG. 2, one half of the die is completely covered with the element 6, and thus the die is biased by the serial arrangement. However, after a computer applies the first placement technique to the dodecahedron, an unbiased numbering may be attained, which is shown in FIG. 3. FIG. 3 shows that the biased set of elements is evenly distributed among the faces of the dodecahedron die in an unbiased manner.

The first placement technique used by a computer determines the placement of elements on the inherently biased die when the c_(i)=1 for the unbiased set of elements. A second and third placement techniques apply to the situation where c_(i)>1 for i=(1 to (n−1)).

Whether a computer applies the second placement technique, which is carried out by a placement module executed by a processor and stored on a non-transitory computer readable medium, or the third placement technique, which is carried out by a placement module executed. by a processor and stored on a non-transitory computer readable medium, depends on an initial determination of whether the sum of c_(i) for i=(1 to (n−1)) greater than c_(i) for i=NOT (1 to (n−1)).

As a second example, an eighteen sided die will be analyzed, where N=18, n=6, and i=1, 2, 3, 4, 5, 6. In this example, c_(i)=2 for elements (1 to (n−1)), which, in this example, are elements 1-5, and c_(i)=8 for i=NOT (1 to (n−1)), which, in this example, is element 6.

Because c_(i)>1, a computer makes an initial determination. Using the initial determination, if a computer determines that the sum of c_(i) for i=(1 to (n−1)) is greater than c_(i) for i=NOT (1 to (n−1)), the unbiased placement of elements on the die is accomplished by placing ((N/n)−2) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements, which is described herein as a second exemplary placement technique. If the c_(i) for i=(1 to (n−1)) is less than or equal to c_(i) for i=NOT (1 to (n−1)), the unbiased placement of elements on the die is accomplished by placing an element from the unbiased set of elements after every (N−3n)/n occurrences an element from the biased set of elements, which is described herein as a third exemplary placement technique.

Returning to the second example, the unbiased set includes the following elements: 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6; and the biased set includes the following elements: 6, 6, 6, 6, 6, 6, 6, 6. The sum of c for i=(1 to (n−1))=10, and c₆=8. 10>8, so the unbiased placement of elements on the die is accomplished by placing the ((N/n)−2) elements from the biased set after every (N−n)/n occurrences of an element from the unbiased set. In this example, ((N/n)−2)=1 and (N−n)/n=2. Thus, 1 element from the biased set is placed after every 2 occurrences of an element from the unbiased set of elements, and the proper order for the elements would be 1, 2, 6, 3, 4, 6, 5, 6, 6, 1, 2, 6, 3, 4, 6, 5, 6, 6,

In a third example, where again c_(i)=2 for i=(1 to (n−1)), if N=24, n=6, sum of c for i=(1 to (n−1))=10, and c₆=14. In this case 10<14, and the unbiased placement by a computer of elements on the die is accomplished by placing an element from the unbiased set of elements after every (N−3n)/n occurrences an element from the biased set of elements. The unbiased set includes 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6; and the biased set includes 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6. By applying the third placement technique, 1 element from the unbiased set after every 1 occurrence of an element from the biased set results in: 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6.

The second and third exemplary placement techniques may be applied by a computer to any value of c_(i) for i=(1 to (n−1)) that is greater than 1. The values in the mathematical formulas of the second and third placement techniques depend upon two value of c_(i) for elements (1 to (n−1)). In every value of c_(i), it must first be determined whether the sum of c_(i) for i=(1 to (n−1)) is greater than c_(i) for i=NOT (1 to (n−1)), as described above. If the sum of c_(i) for i=(1 to (n−1)) is greater than c_(i) for i=NOT (1 to (n−1)), then the unbiased placement of elements on the die is accomplished by placing ((N/n)−c_(i)) elements from the biased set of elements after every (N−n)/n occurrences of an element from the unbiased set of elements. If the sum of c for i=(1 to (n−1)) is less than or equal to c for i=NOT (1 to (n−1)), then the unbiased placement of elements on the die is accomplished by placing an element from the unbiased set of elements after every (N−((c_(i)+1*n))/n occurrences an element from the biased set,

As a fourth example, N=24, n=6, and i=1, 2, 3, 4, 5, 6. For this example, let c_(i)=3 for elements (1 to (n−1)), which, in this example, are elements 1-5, and c_(i)=9 for i NOT (1 to (n−1)), which, in this example, is element 6.

In this fourth example, the unbiased set includes the following elements: 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6; and the biased set includes the following elements: 6, 6, 6, 6, 6, 6. The sum of c for i=(1 to (n−1))=15, and c₆=9. 15>9, so the second placement technique is used, and the unbiased placement of elements on the die is accomplished by a computer placing ((N/n)−3) elements from the biased set after ever (N−n)/n occurrences of an element from the unbiased set. In this example, ((N/n)−3)=1 and (N−n)/n=3. Thus, 1 element from the biased set is placed after every 3 occurrences of an element from the unbiased set. As a result, the proper order for the elements would be 1, 2, 3, 6, 4, 5, 6, 6, 1, 2, 3, 6, 4, 5, 6, 6, 1, 2, 3, 6, 4, 5, 6, 6.

In a fifth example, where again c_(i)=3 for i=(1 to (n−1)), if N=30, n=6, sum of c_(i) for i=(1 to (n−1))=15, and c₆=15. In this case 15=15, so the third placement technique is used, and the unbiased placement of elements on the die is accomplished by placing an element from the unbiased set after every (N−4n)/n occurrences an element from the biased set. Thus, the elements 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6 may place 1 element from the unbiased set after every (N−4n)/n occurrences of an element from the biased set. The unbiased set includes 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6; and the biased set includes 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6. (N−4n)/n=1 so a computer places 1 element from the unbiased set after every 1 occurrence of an element from the biased set results in: 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 6.

A fourth exemplary placement technique, which is carried out by a placement module executed by a processor and stored on a non-transitory computer readable medium, for unbiased placement of elements on an inherently biased die can be applied to the situation where c_(i)=n for i=(1 to (n−1)).

To illustrate the fourth placement technique, a 42-sided die (N=42) having six unique elements (n=6) is taken as an example. The six unique elements are i=1, 2, 3, 4, 5, 6, and, c_(i)=6 for i=1, 2, 3, 4, 5, and c_(i)=12 for i=6. Because c_(i)=n, the fourth placement technique is implemented. In the fourth placement technique, (N/n−n) elements from the biased set of elements is placed after every (N−n)/n occurrences of an element from the set of unbiased elements. Applying the first placement technique to the example above, all of the elements on the die would be 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6. The unbiased set would be 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6; and the biased set would be 6, 6, 6, 6, 6, 6. By having a computer apply the mathematical model used by the fourth placement technique, (N−n)/n=6, and ((N/n)−n)=1, and thus, 1 element from the biased set is placed after every 6 occurrence of an element from the unbiased set. Thus, the placement of the elements on the die is in the order of 1, 2, 3, 4, 5, 6, 6, 1, 2, 3, 4, 5, 6, 6, 2, 3, 4, 5, 6, 6, 1, 2, 3, 4, 5, 6, 6, 1, 2, 3, 4, 5, 6, 6, 1, 2, 3, 4, 5, 6, 6,

As a final example, FIGS. 4-5 illustrate a 20-sided die. However, in this example, the 20-sided die has live elements (n=5), such that i=1, 2, 3, 4, 5. In this example, c_(i)=3 for elements i (1 to (n−1)), which, in this example, are elements 1, 2, 4 and 5, and for element i=3, c_(i)=8. So, the sum of c_(i) for i=(1 to (n−1)) is equal to 12and the sum of c₃ is 8. Thus, the second placement technique is used, and the formulas (N/n−c_(i)) and (N−n)/n are used, and as a result (N/n−c_(i))=1 and ((N−n)/n)=3.

The unbiased element set includes 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 3, 4, 5; and the biased set includes 3, 3, 3, 3, 3. When a computer uses the second placement technique, one element from the biased set is placed after every three occurrences of an element from the unbiased set, and the result is: 1, 2, 3, 3, 4, 5, 1, 3, 2, 3, 4, 3, 5, 2, 3, 3, 4, 5, 3. FIG. 4 illustrates a biased placement of elements on a icosahedron, and FIG. 5 illustrates an unbiased placement of elements on the same icosahedron for reference.

An exemplary method of determining which placement technique to apply is illustrated in FIG. 6. This exemplary method is implemented by a specially programmed computer, such as a computer program product on a non-transient computer-readable medium that is executed by a processor to perform this method. The method begins in step 610, where it is first determined if the count c_(i)=1 for the unbiased set of elements. If the count is one, the first placement technique is applied in step 620. If the count for the unbiased set of elements if greater than one, the method continues to step 622 where it is determined whether c_(i)=n for the unbiased set of elements. If the c_(i)=n, the fourth placement technique is applied in step 624. If the count of the for the unbiased set of elements if greater than one and less than n, the method continues to step 630 where it is determined whether the sum of c_(i) for i=(1 to (n−1)) is greater than c_(i) for i=NOT (1 to (n−1)). If the answer to the determination in step 630 is yes, the method continues to step 640, and the second placement technique is implemented. If the answer to the determination in step 630 is no, the method continues to step 650, and the third placement technique is implemented.

The exemplary placement techniques described herein can be performed by a computer that places and arranges elements in a random and unbiased order on an inherently biased die. The placement techniques may be applied by a computer to a die of any number of sides, as long as the number of sides (N) is exactly divisible by the number of unique elements (n) to be included on the faces of the die and [N<=(2n+n)].

Several additional examples of different dice with differing numbers of faces and differing numbers of elements to place on the faces, FIGS. 7A-C illustrate examples where c_(i)=1, FIGS. 8A-8C illustrate examples where c_(i)=2, FIGS. 9A-C illustrate examples where c_(i)=3, FIGS. 10A-C illustrate examples where c_(i)=4, FIGS. 11A-C illustrate examples where c_(i)=5, FIGS. 12A-C illustrate examples where c_(i)=6. In FIGS. 7-12, one star (*) next to a number indicates that the summation of c for (1 to (n−1))<than summation c for NOT (1−(n−)), and two stars (**) next to a number indicates that the summation of c for (1 to (n−1))>=than summation c for NOT (1−(n−)). The examples is FIGS. 7-12 are continuing the same exemplary format as above in tabular form.

The exemplary embodiments can include one or more computer programs that embody the functions described herein and illustrated in the appended flow charts. However, it should be apparent that there could be many different ways of implementing aspects of the exemplary embodiments in computer programming, and these aspects should not be construed as limited to one set of computer instructions. Further, those skilled in the art will appreciate that one or more acts described herein may be performed by hardware, software, or a combination thereof, as may be embodied in one or more computing systems.

The functionality described herein can be implemented by numerous modules or components that can perform one or multiple functions. Each module or component can be executed by a computer, such as a server, having a non-transitory computer-readable medium and processor. In one alternative, multiple computers may be necessary to implement the functionality of one module or component.

Unless specifically stated otherwise as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as “processing” or “computing” or “calculating” or “determining” or “displaying” or “generating” or “synchronizing” or “outputting” or the like, can refer to the action and processes of a data processing system, or similar electronic device, that manipulates and transforms data represented. as physical (electronic) quantities within the system's registers and memories into other data similarly represented as physical quantities within the system's memories or registers or other such information storage, transmission or display devices.

The exemplary embodiments can relate to an apparatus for performing one or more of the functions described herein. This apparatus may be specially constructed for the required purposes, or it may comprise a general purpose computer selectively activated or reconfigured by a computer program stored in the computer. Such a computer program may be stored in a machine (e.g. computer) readable storage medium, such as, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs and magnetic-optical disks, read only memories (ROMs), random access memories (RAMs) erasable programmable ROMs (EPROMs), electrically erasable programmable ROMs (EEPROMs), magnetic or optical cards, or any type of media suitable for storing electronic instructions, and each coupled to a bus.

The exemplary embodiments described herein are described as software executed on at least one server, though it is understood that embodiments can be configured in other ways and retain functionality. The embodiments can be implemented on known devices such as a personal computer, a special purpose computer, cellular telephone, personal digital assistant (“PDA”), a digital camera, a digital tablet, an electronic gaming system, a programmed microprocessor or microcontroller and peripheral integrated circuit element(s), and ASIC or other integrated circuit, a digital signal processor, a hard-wired electronic or logic circuit such as a discrete element circuit, a programmable logic device such as a PLD, PLA, FPGA, PAL, or the like. In general, any device capable of implementing the processes described herein can be used to implement the systems and techniques according to this invention.

It is to be appreciated that the various components of the technology can be located at distant portions of a distributed network and/or the Internet, or within a dedicated secure, unsecured and/or encrypted system. Thus, it should be appreciated that the components of the system can be combined into one or more devices or co-located on a particular node of a distributed network, such as a telecommunications network. As will be appreciated from the description, and for reasons of computational efficiency, the components of the system can be arranged at any location within a distributed network without affecting the operation of the system. Moreover, the components could be embedded in a dedicated machine.

Furthermore, it should be appreciated that the various links connecting the elements can be wired or wireless links, or any combination thereof, or any other known or later developed element(s) that is capable of supplying and/or communicating data to and from the connected elements. The term module as used herein can refer to any known or later developed hardware, software, firmware, or combination thereof that is capable of performing the functionality associated with that element. The terms determine, calculate and compute, and variations thereof, as used herein are used interchangeably and include any type of methodology, process, mathematical operation or technique.

The embodiments described above are intended to be exemplary. One skilled in the art recognizes that numerous alternative components and embodiments that may be substituted for the particular examples described herein and still fall within the scope of the invention. 

What is claimed is:
 1. A die comprising: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, when a count (c_(i)) of each unique element in the unbiased set is equal to
 1. 2. The die of claim 1, wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased.
 3. A die comprising: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing ((N/n)−c_(i)) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements, if a count of the elements in the unbiased set less the elements having an identical value as the elements in the biased set of elements is greater than a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set.
 4. The die of claim 3, wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased.
 5. A die comprising: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−((c_(i)+1)*n))/n occurrences an element from the biased set of elements, if a count of the elements in the unbiased set less the elements having an identical value as the elements in the biased set of elements is less than or equal to a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set.
 6. The die of claim 3, wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased.
 7. A die comprising: a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing ((N/n)−n) elements from the biased set of elements after every (N−n)/n occurrences of an element from the unbiased set of elements, when a count (c_(i)) of each unique element in the unbiased set is equal to n.
 8. The die of claim 7, wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased.
 9. A computer-implemented method of labeling elements on an inherently biased die having a plurality of faces (N), comprising: determining, by a computer, whether the count of each unique element (c_(i)) in an unbiased set of elements is greater than 1; applying, computer, a first placement means if the c_(i) of the unbiased set of elements is equal to one; determining, by a computer, whether the count of each unique element (c_(i)) in an unbiased set of elements is equal to a number of elements (n); applying, by a computer, a fourth placement means if the c_(i) of the unbiased set of elements is equal to n; determining, by a computer, whether a count of the elements in the unbiased set less the elements having an identical value as the elements in an biased set of elements (c1) is greater than a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set (c2); applying, by a computer, a second placement means if c1is greater than c2; and applying, by a computer, a third placement means if c1 is less than or equal to c2.
 10. The method of claim 9, wherein the first placement means comprises a first placement module that places an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, where N is the number of faces on the die and n is the number of unique elements included on the die.
 11. The method of claim 9, wherein the second placement means comprises a second placement module that places ((N/n)−c_(i)) elements from the biased set of elements after ever n)/n occurrences of an element from the unbiased set of elements, where N is the number of faces on the die and n is the number of unique elements included on the die.
 12. The method of claim 9, wherein the third placement means comprises a third placement module that places an element from the unbiased set of elements after every (N−((c_(i)+1)*n))/n occurrences of an element from the biased set of elements, where N is the number of faces on the die and n is the number of unique elements included on the die.
 13. The method of claim 9, wherein the fourth placement means comprises a fourth placement module that places ((N/n)−n) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements.
 14. The method of claim 9, wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased.
 15. The method of claim 9, wherein the N is exactly divisible by n, and [N<=(n2+n)]. 